Subspace method based on neural networks for eigenvalue problems

TL;DR

This paper introduces a neural network-based subspace method for eigenvalue problems that achieves high accuracy and low computational cost by constructing orthogonal bases through deep learning and dimensionality reduction.

Contribution

It presents a novel approach combining neural networks and subspace methods to efficiently solve eigenvalue problems with improved accuracy and reduced computational effort.

Findings

High-accuracy eigenvalues and eigenfunctions obtained

Method reduces computational cost significantly

Numerical experiments validate effectiveness

Abstract

In this paper, we propose a subspace method based on neural networks for eigenvalue problems with high accuracy and low cost. We first construct a neural network-based orthogonal basis by some deep learning method and dimensionality reduction technique, and then calculate the Galerkin projection of the eigenvalue problem onto the subspace spanned by the orthogonal basis and obtain an approximate solution. Numerical experiments show that we can obtain approximate eigenvalues and eigenfunctions with very high accuracy but low cost.